Fitness Proportionate Selection with Roulette Wheel and Stochastic Universal Sampling

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Holland's original GA used fitness-proportionate selection, in which the "expected value" of an individual (i.e., the expected number of times an individual will be selected to reproduce) is that individual's fitness divided by the average fitness of the population. The most common method for implementing this is "roulette wheel" sampling, described in chapter 1: each individual is assigned a slice of a circular "roulette wheel," the size of the slice being proportional to the individual's fitness. The wheel is spun N times, where N is the number of individuals in the population. On each spin, the individual under the wheel's marker is selected to be in the pool of parents for the next generation. This method can be implemented as follows:

Sum the total expected value of individuals in the population. Call this sum T.

Repeat N times:

Choose a random integer r between 0 and T.

Loop through the individuals in the population, summing the expected values, until the sum is greater than or equal to r. The individual whose expected value puts the sum over this limit is the one selected.

This stochastic method statistically results in the expected number of offspring for each individual. However, with the relatively small populations typically used in GAs, the actual number of offspring allocated to an individual is often far from its expected value (an extremely unlikely series of spins of the roulette wheel could even allocate all offspring to the worst individual in the population). James Baker (1987) proposed a different sampling method—"stochastic universal sampling" (SUS)—to minimize this "spread" (the range of possible actual values, given an expected value). Rather than spin the roulette wheel N times to select N parents, SUS spins the wheel once—but with N equally spaced pointers, which are used to selected the N parents. Baker (1987) gives the following code fragment for SUS (in C):

ptr = Rand(); /* Returns random number uniformly distributed in [0,1] */ for (sum = i = 0; i < N; i++)

for (sum += ExpVal(i,t); sum > ptr; ptr++) Select(i);

where i is an index over population members and where ExpVal(i,t) gives the expected value of individual i at time t. Under this method, each individual i is guaranteed to reproduce at least ExpVal(i,t) times but no more than ExpVal(i,t) times. (The proof of this is left as an exercise.)

SUS does not solve the major problems with fitness-proportionate selection. Typically, early in the search the fitness variance in the population is high and a small number of individuals are much fitter than the others. Under fitness-proportionate selection, they and their descendents will multiply quickly in the population, in effect preventing the GA from doing any further exploration. This is known as "premature convergence." In other words, fitness-proportionate selection early on often puts too much emphasis on "exploitation" of highly fit strings at the expense of exploration of other regions of the search space. Later in the search, when all individuals in the population are very similar (the fitness variance is low), there are no real fitness differences for selection to exploit, and evolution grinds to a near halt. Thus, the rate of evolution depends on the variance of fitnesses in the population.

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